An operator representation for weighted spaces of vector valued homomorphic functions
type: Item
instance of: scholarly article
title: An operator representation for weighted spaces of vector valued homomorphic functions
zbMATH Open document ID: 0942.46023
DOI: 10.1007/BF03322097
author: Silke Holtmanns
author: Klaus Dieter Bierstedt
published in: Results in Mathematics
publication date: 2000-08-23T00:00:00Z
review text: Let \(G\) be an open set in \(\mathbb{C}^N\) and \(V\) a Nachbin family of weights on \(G\) such that for each compact set \(K\) in \(G\) there exists \(v\in V\) with \(\inf_{z\in K} v(z)>0\). For a locally convex (l.c.) space \(E\) define the associated weighted space of holomorphic functions as \[ HV(G,E):= \{f: G\to E:f\text{ is holomorphic and \(vf\) is bounded in \(E\;\forall v\in V\)}\}, \] and endow it with the l.c. topology induced by the semi-norms \[ q_{v,p}: f\mapsto \sup\{v(z)p(f(z)): z\in G\},\;p\text{ a continuous seminorm on }E. \] Then the main results of the paper are: For each quasibarrelled l.c. space \(E\) the map \[ \Phi: HV(G, E_b')\to{\mathcal L}_b(E, HV(G,\mathbb{C})),\;\Phi(f): e\mapsto f(\cdot)[e] \] is a linear topological isomorphism. Furthermore, if \(G_j\subset \mathbb{C}^{N_j}\) and \(V_j\) are as above for \(j= 1,2\), then \[ HV_1(G_1, HV_2(G_2))= H(V_1\otimes V_2)(G_1\times G_2). \] Besides this, related results for continuous functions, applications and connections to work of other authors are discussed.
reviewed by: Reinhold Meise
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46A20
Mathematics Subject Classification ID: 46A32
zbMATH DE Number: 1346909
zbMATH Keywords: quasibarrelled space
zbMATH Keywords: weighted space of holomorphic functions
zbMATH Keywords: linear topological isomorphism
MaRDI profile type: Publication
cites work: Dual density conditions in (DF)-spaces. I
cites work: Q4015688
cites work: Weighted spaces of holomorphic functions on balanced domains
cites work: Approximationseigenschaft, Lifting und Kohomologie bei lokalkonvexen Produktgarben
cites work: A Projective Description of Weighted Inductive Limits
cites work: Vitushkin's localization operator and holomorphic functions in polydiscs
cites work: Q3748742
cites work: Q5819711
cites work: Q5833442
cites work: Q3918541
cites work: Q5561411
cites work: Q4282711
full work available at URL: https://doi.org/10.1007/bf03322097
OpenAlex ID: W2008690767
description: scientific article; zbMATH DE number 1346909